A formal proof of this theorem can be found in Appendix REF .
We note that our proof further simplifies derivations using the separability of the mirror map.
The final missing piece would be the existence of the regularized direction.
In general, finding the limit direction \(u^{\sf \tiny r}_{p}\)  would be difficult.
Fortunately, we can sometimes appeal to the max-margin direction that is much easier to compute.
The following result is a generalization of [1]} and shows that for common losses in classification, the regularized direction and the max-margin direction are the same, hence proving the existence of the former.
